I cannot watch youtube or other flash videos on fullscreen on the external monitor plugged into my laptop and type on the other screen, since the fullscreen video will go back to small screen if I try typing on the other screen. Is there a way to fix this?

The Flash plugin is specifically designed to un-fullscreen as soon as the video loses focus. As far as I am aware there is no way to change that, nor is there a way to hide the "Press ESC to leave full screen" message that shows up upon entering fullscreen mode.